Finite element modeling of thermal induced fracture ...

$: Finite element modeling of thermal induced fracture ...$
POLITECNICO DI MILANO

in partial fulfillment of the requirements for the degree of

Doctor in Structural, Earthquake and Geotechnical Engineering

XXIV cycle

Finite element modeling of thermal induced

fracture propagation in brittle materials

by

Giovanna Bucci

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Roberto Paolucci

Doctoral School of Structural, Earthquake and GeotechnicalEngineeringCoordinator

March 2012

2

Acknowledgments

All my gratitude

to Professor Anna Pandolfi for introducing me to the nonlinear solid mechanics and

its finite element code implementation

to Progetto Roberto Rocca, for investing in my education and offering me invaluable

opportunities in the US

to Professor Raul Radovitzky, for welcoming me in his research group at MIT,

where my curiosity has been encouraged and nourished

to my mentor Professor Carlo Cinquini, for always being a reference and a model

alla mia famiglia, perche conosce tutta la storia dall’inizio

to Bart, Gail and Greg, for being my American family

to Brandon, for all the rest

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Contents

1 Introduction 11

1.1 Inspiring experimental results . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Subject overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3 Analysis and modeling of the problem . . . . . . . . . . . . . . . . . . 22

1.3.1 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . 22

1.3.2 Two-dimensional modelling of the heat problem . . . . . . . . 26

1.3.3 Steady thermal profile . . . . . . . . . . . . . . . . . . . . . . 27

1.3.4 Organization of the contents . . . . . . . . . . . . . . . . . . . 28

2 Non-linear Solid Mechanics 29

2.1 Thermodynamics of non-linear elastic materials . . . . . . . . . . . . 29

2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.1.2 Thermodynamic processes . . . . . . . . . . . . . . . . . . . . 30

2.1.3 Constitutive equations . . . . . . . . . . . . . . . . . . . . . . 32

2.1.4 Entropy inequality principle . . . . . . . . . . . . . . . . . . . 33

2.1.5 Thermoelastic field equations . . . . . . . . . . . . . . . . . . 35

2.1.6 State functions for thermo-elastic materials . . . . . . . . . . . 37

2.1.7 Constitutive law: Isotropic Hyperelastic Model . . . . . . . . . 38

2.2 Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2.1 Irreversible Cohesive Fracture model in a Non-linear kinematics

framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5

3 Numerical procedure 47

3.1 Heat conduction problem . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Weak form and discretization of the mechanical problem . . . . . . . 48

3.2.1 Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.2 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . 50

3.3 Dynamic Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.1 Integration parameters . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4.1 Finite element implementation of fracture . . . . . . . . . . . 56

3.5 Synthesis of the numerical strategy . . . . . . . . . . . . . . . . . . . 59

4 Numerical results 61

4.1 Numerical test setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Qualitative discussion on the crack patterns resulting from the simu-

lations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1 Fracture propagation conditioned by mesh dependency. . . . . 66

4.2.2 Crack morphology dependence on the immersion velocity. . . . 66

4.2.3 Crack pattern depending on the plate width. . . . . . . . . . . 69

4.3 Considerations about crack tip position and energy dissipation in the

fracture process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Analytical consideration on the instability manifested in the crack

propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Conclusion 81

5.1 Current development . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Possible Further Developments . . . . . . . . . . . . . . . . . . . . . . 83

A Properties and invariants of kinematic variables 85

A.0.1 Index notation . . . . . . . . . . . . . . . . . . . . . . . . . . 85

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List of Figures

1-1 Schematic illustration of the set-up for the numerical tests we aim to

perform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1-2 Crack patterns recorded in [43], [44]: a) straight propagation; b) dumped

oscillating pattern; c) sinusoidal pattern; d) oscillating propagation in

a large plate; e) oscillating crack propagating along a narrow plate. . 15

1-3 Instable crack patterns documented in [43], [44] . . . . . . . . . . . . 17

1-4 The data resulting from the experiments on glass plates (of dimensions

24 x 60 x 0.13 mm3 carried out by Yuse and Sano have been grouped

in a phase diagram in relation to the values of immersion velocity and

temperature gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1-5 Three states are observed according to the sample width W : a) no

propagation for W < Wc; b)straight propagation for Wc < W < Wosc;

c)oscillating propagation for W > Wosc; e),d)non-regular oscillation for

W >> Wosc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1-6 Glass plate characterized by increasing width. The sample has been

used by [34] to estimate the critical value Wosc, which corresponds to

the de-stabilization of the straight crack. . . . . . . . . . . . . . . . . 20

1-7 Helical cracks in cylindrical samples quenched in cold water. Exper-

iment recorded in [44] to underline the effect of different boundary

conditions on the crack path. . . . . . . . . . . . . . . . . . . . . . . 21

1-8 Geometry of the experimental setup. . . . . . . . . . . . . . . . . . . 25

2-1 Cohesive surface traversing a 3D body . . . . . . . . . . . . . . . . . 41

7

2-2 Adopted cohesive law expressed in terms of an effective opening dis-

placement δ and a traction t: loading-unloading rule from linearly

decreasing loading envelop. . . . . . . . . . . . . . . . . . . . . . . . . 44

3-1 Geometry of tringular cohesive element inserted between two 10-node

tetrahedral elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3-2 (a) Standard element configuration and natural co-ordinate system; (b)

deformed middle surface S and corresponding curvilinear coordinate

system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3-3 Opening displacement . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3-4 Flow chart of the numerical procedure for the solution of the thermo-

mechanical problem and the detection of fracture propagation. . . . . 59

4-1 Phase Field of the crack morphology, depending on the parameters: V

and ∆ϑ. The value κ/H represents the limit of validity of the steady

thermal state assumption. . . . . . . . . . . . . . . . . . . . . . . . . 63

4-2 Fracture patterns resulting from ∆ϑ = 350 C. In correspondence of

each figure the caption expresses the velocity in mms−1. . . . . . . . . 64









4-7 Fracture patterns resulting from ∆ϑ = 250 C . . . . . . . . . . . . . 67

8

4-8 Experimental results of [34] relate the crack morphology to the plate

width and the driving velocity V . The treshold lines reflect the vari-

ation of the thermal field with the V . The transition lines are con-

trolled by (a) the distance H between the reservoirs at low velocities;

(b) the thermal diffusion lenght dth in the decreasing part of the graph;

(c) the thickness of the plate for higher velocities, leading to a three-

dimensional fracture problem and a non-smooth crack surface. . . . . 68

4-9 Phase Field of the crack morphology, depending on the dimensionless

parameters: Pe and τ . The value Pe = 1 represents the limit of

validity of the steady thermal state assumption. . . . . . . . . . . . . 69

4-10 Fracture patterns resulting from ∆ϑ = 350 C and V = 0.001mms−1

(Cauchy stress units: N/mm2). . . . . . . . . . . . . . . . . . . . . . 70

4-11 Fracture patterns resulting from ∆ϑ = 300 C and V = 0.0001mms−1

(Cauchy stress units: N/mm2). . . . . . . . . . . . . . . . . . . . . . 71

4-12 ∆ϑ = 150 C, V = 0.0001mm/s . . . . . . . . . . . . . . . . . . . . . 73

4-13 ∆ϑ = 200 C, V = 0.0001mms−1 . . . . . . . . . . . . . . . . . . . . . 73

4-14 ∆ϑ = 300 C, V = 0.0001mms−1 . . . . . . . . . . . . . . . . . . . . . 73

4-15 ∆ϑ = 350 C, V = 0.0005mm/s . . . . . . . . . . . . . . . . . . . . . 74

4-16 ∆ϑ = 250 C, V = 0.0001mms−1, refined mesh . . . . . . . . . . . . . 74

4-17 ∆ϑ = 250 C, V = 0.00025mms−1, refined mesh . . . . . . . . . . . . 75

4-18 ∆ϑ = 250 C, V = 0.0005mms−1, refined mesh . . . . . . . . . . . . . 75

4-19 Simple experiment of a hoop rotating about its vertical axis. The

position occupied by the ball depends on the angular velocity of the

rotation. When a critical value w0 is reached, the configuration with

the ball at the bottom of the hoop becomes unstable. . . . . . . . . . 77

4-20 Reference system for the description of the crack tip position respect

to the water bath surface, denoted by the parameter s. . . . . . . . . 79

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Chapter 1

Introduction

The development of a broad understanding of fracture has embraced the underlying

processes determining the deformation and strength of the material as well as the

influences of the environment upon them. Researches have been concentrated on

studying the microscopic mechanisms which govern fracture and on the establishment,

from a macroscopic point of view, of a quantitative fracture criteria for assessing

fracture of components.

According to Griffith’s theory [15] the quasi-static propagation of a brittle crack

is governed by the balance between energy release rate and fracture toughness

G = Gc. (1.1)

The driving force for crack propagation is the so called energy release rate G , the

amount of elastic energy released by the body when the crack itself advances of a

unit area. It is function of geometry and load, instead Gc is regarded as a material

parameter.

This concept has been the subject of several fundamental works in fracture me-

chanics among which a couple of landmarks should be cited: the representation of G

in terms of the stress intensity factors, due to Irwin [21], and the representation in

terms of the J-integral, due to Rice [31]. Through the Irwin formula the interpretation

of the fracture propagation criterion may be shifted from the energy based condition

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to one based on stress intensity. In perfectly brittle materials we may write

KI = KIc, (1.2)

where the stress intensity factor KI is expression of the stress concentration at the

crack tip under mode I load and the fracture toughness KIc is a material property.

The nucleation or the condition of growing are non sufficient information to describe

the fracture process. The path along which a crack propagates when the body is

under mixed mode loading is, in fact, generally non-straight. The prediction of the

crack path has represented a classical challenge for the fracture mechanics, several are

the contributions to the literature, starting from the ’60. Among them the principal

of local symmetry [14], maximum energy release rate [5] and the maximum circum-

ferential stress [37] are able to predict the angle of deviation of the crack (kinking

angle). Classical analytical results on the principal of local symmetry are provided

by mean of an asymptotic expansion of the crack path in a small neighborhood of the

kinking point. For instance the governing equation of the principle of local symmetry

is

KII = 0, (1.3)

i.e., that the crack always propagates in mode I, so that in-plane tractions that remain

perpendicular to the crack in a small neighborhood of the crack tip. These analytical

methods are able to detect the instability of the crack respect to the straight propa-

gation within some geometric and loading restrictions. The aim of this research is to

give a contribution to the tracking of complex fracture patterns without some of the

limitations imposed by analytical formulations and exploiting part of the potential

of the numerical treatment of the problem. We focus on brittle fracture propagation

in quasi-static condition and in presence of a thermal gradient which represents the

driving loading of the crack growth. A transient thermal gradient applied to a con-

ductive body acts as a mixed mode loading and provides less explored conditions for

the study of fracture mechanics. At the same time heating and/or cooling may be a

reason of damaging and crack nucleation, particularly in thin structures. Our goal is

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to develop a finite element model in order to describe the thermo-mechanical process

and the fracture phenomenon in thin plates made of brittle materials, such as glass

or PMMA.

In a local characterization of the mechanisms of fracture, we conveniently define a

fracture cohesive zone. In a small region ahead of the crack tip the separation process

is supposed to be active but not complete: the two fracture flanks interact with forces

that decrease with the opening displacements, as long as the distance which separate

the crack lips does not reach a critical value. The concept of cohesive zone model is due

to Dugdale [10] and Barenblatt [1] while an important contribution for its numerical

implementation by mean of cohesive finite elements was offered by Camacho-Ortiz [4]

and Pandolfi-Ortiz [29] [28]. This technique allows the representation of fracture as a

discontinuity in the domain and it may be a powerful tool to follow the crack growth

without requiring any a-priori knowledge about its path. One of our main interest is,

in fact, to capture the origin of possible instabilities in the fracture direction, in the

form of kinking and branching, the appearance of patterns characterized by different

level of regularity, symmetry and repetitiveness.

1.1 Inspiring experimental results

The propagation direction of a crack as a function of the external loading and in

particular the stability of a straight propagation are fundamental problems of fracture

mechanics but not yet fully understood. One of the obstacles to this understanding

is represented by the difficulty to perform well-controlled experiments.

Multiple simple tests have showed that a crack travelling in a thin glass plate

with a thermal stress field undergoes a reproducible sequence of instabilities. The

experiment (sketched in Fig.1-1) is conducted by slowly pulling the sample from a

hot region to a cold one, so that in effect a sharp thermal gradient moves across the

plate. The plate is seeded with a crack , which the thermal stresses cause to extend

at the immersion speed. Since the adopted range of velocities is much slower than the

speed of sound in glass, the phenomenon may be regarded as quasi-static. Fracture

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Figure 1-1: Schematic illustration of the set-up for the numerical tests we aim toperform

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propagation in the glass plate due to sudden but controlled cooling shows drastic

morphological changes depending on the values of the experimental parameters. The

patterns observed can be grouped in four categories:

i) no propagation

ii) straight propagation (see Fig.1-2a)

iii) crack oscillating with different possible amplitudes (see Fig.1-2e, Fig.1-2c, Fig.1-

2d, Fig1-3a)

iv) branching patterns of various complexity (see Fig.1-3b, Fig.1-3c)

(a) (b) (c) (d) (e)

Figure 1-2: Crack patterns recorded in [43], [44]: a) straight propagation; b) dumpedoscillating pattern; c) sinusoidal pattern; d) oscillating propagation in a large plate; e)oscillating crack propagating along a narrow plate.

The amorphous nature of glass is supposed to not interfere with the direction of

fracture kinking. A similar experimental set, adopted for single-crystal silicon wafers

by Deegan et al. [9], has revealed similar instabilities in the fracture development,

15

even starting from an asymmetric notch and accounting for the relevant anisotropy

of the material.

The nucleation of a crack in absence of an initial notch requires in general a severe

temperature gradient, so a large value of ∆ϑ, the temperature difference between hot

and cold reservoir. In order to perform experiments with a limited temperature gap

and avoid many fractures spontaneously propagating, according to the experimen-

talists the sample has been equipped with an initial vertical crack in the center of

the bottom edge of the plate. The initial crack may perturb the developing pattern

but after a transient the effect tends to disappear re-establishing the conditions due

to the control parameters. If the orientation of the notch is inclined respect to the

vertical axis, the propagation shows an initial oscillation dumped to a straight line

(see Fig.1-2b), when the experiment is performed within the range of straight crack

conditions.

Each couple of values (∆ϑ, V ), where V is the immersion velocity, provides the

system with a certain amount of internal energy available for fracturing the material.

The thermal gradient and the imposed velocity need to be severe enough to drive the

propagation of the crack. During growth the crack may select different patterns, to

dissipate different quantities of energy by creating of new surfaces. Increasing the

value of ∆ϑ, for a fixed driving velocity, it is possible to observe the transition from

straight to periodic oscillating propagation and finally to the branched one, in the

sense of a progressive extension of the total fracture area.

The wavelength and the amplitude characterizing the oscillation in the fracture

profiles depend on the control parameters. When the velocity is slightly beyond

the onset of oscillation, the crack shape is almost sinusoidal (see Fig.1-2c). As V

or ∆ϑ is increased, the shape becomes a sequence of semicircles (see Fig.1-2d) and

finally becomes asymmetric in the direction of propagation (see Fig.1-3a). Oscillations

of branched cracks could be irregular, their amplitude modulated or chaotic. The

number of branches varies with the driving parameters, different levels of complexity

and disorder have been observed. Yuse and Sano [43], [44] investigated the role of the

descent velocity V and of the temperature gap between the initial plate temperature

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(a) oscillating crackpattern describingsemicircles

(b) two-branchcrack withoscillation

(c) pattern with mul-tiple branches

Figure 1-3: Instable crack patterns documented in [43], [44]

17

Figure 1-4: The data resulting from the experiments on glass plates (of dimensions 24x 60 x 0.13 mm3 carried out by Yuse and Sano have been grouped in a phase diagramin relation to the values of immersion velocity and temperature gap.

and the cold reservoir ∆ϑ, provided by a water bath.

The upper dashed line in 1-4 denotes the threshold for the appearance for branched

cracks, with some dispersion. The lower solid line represents the boundary between

the oscillating pattern and the straight one. Threshold lines have negative slope for

V < 10mms−1 and positive slope for V > 10mms−1. This result has been interpreted

by the authors as due to the cooling delay of the last portion of the plate immersed

into the water (heat transient regime) and to a non isothermal condition across the

sample thickness. The thickness of the plate related to the thermal diffusion length

dth =κ

V, (1.4)

with κ being the glass thermal diffusion coefficient

κ =k

%Cv(1.5)

determines, in fact, the passage from a two- to a three-dimensional processes.

18

In their study Ronsin, Heslot and Perrin [32], [33], [34] introduced a third pa-

rameter represented by the plate width W and they focused on the influence of the

geometry of the plate on the propagation of a single or multiple cracks. For a given

thermal field the internal energy stored in the glass plate depends on specimen width.

The quantity of energy accumulated into the material is released to feed the fracture

advance and to influence the crack pattern. The set of Fig.1-5 illustrates, for a given

thermal field (∆ϑ, the spatial gap between hot and cold region H, V fixed), the crack

growth state dependent on the plate width in relation to some critical values which

represent the transition from no crack development to the straight propagation and

eventually to the oscillating one (see Fig. 1-5). It is also interesting to observe the

progressive destabilization of the pattern respect to the symmetric straight line in a

sample with increasing width (Fig. 1-6). These experimental results are recalled

in more detail in the section 4.2.3 with the purpose of analysing and comparing our

numerical outcomes.

1.2 Subject overview

A straight crack propagating in the middle of the plate is, according to the symmetries,

a pure mode I fracture problem. Crack initiation observations reveal that the presence

of shear stress at the crack tip leads to a direction of growth making a finite angle

with the initial crack direction. This suggests that smooth propagation occurs along

a path where the condition for the crack tip is given by

KII = 0. (1.6)

This is usually referred to as the criterion of local symmetry [14]. Applying this con-

cept, Cotterel and Rice [6] have demonstrated that for a crack growing in an infinite

plate, the stability of the straight direction is due to the non-singular longitudinal

tensile stress T at the crack tip. In presence of a positive value of T , the fracture

deviates from the initial trajectory of motion, instead if T is negative the crack stably

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Figure 1-5: Three states are observed according to the sample width W : a) nopropagation for W < Wc; b)straight propagation for Wc < W < Wosc; c)oscillatingpropagation for W > Wosc; e),d)non-regular oscillation for W >> Wosc

Figure 1-6: Glass plate characterized by increasing width. The sample has been usedby [34] to estimate the critical value Wosc, which corresponds to the de-stabilization ofthe straight crack.

20

Figure 1-7: Helical cracks in cylindrical samples quenched in cold water. Experimentrecorded in [44] to underline the effect of different boundary conditions on the crackpath.

propagates following a straight path. The limit of the application of Cotterel and

Rice condition is related to the sample width W : to determine when the infinite

plate approximation might be appropriate, the dimension W needs to be compared

to the thermal diffusion length. The restriction to large enough values of W and V

may be considered as a limit of the criterion. Furthermore, the boundary conditions

represented by the free lateral edges of the plate are necessary for the oscillation as

well as the instability phenomenon which first deviates the crack form the straight

propagation. The boundary allows the relaxation of the stress near the edges and

induces a restoring force towards the center, where the tensile stress due to the ther-

mal contraction is larger. As a matter of fact, in cylindrical samples no change of

direction has been observed (Fig.1-7) the crack propagates with a fix angle following

an helical pattern whose pitch depends on the experimental parameters.

The transition from a straight to an oscillatory propagation has been interpreted

21

in literature through the Cotterel and Rice criterion by Marder [23] as well as a Hopf

bifurcation by Sasa et al. [36], in both cases referring to infinite plate approximation.

When a problem depends on some parameters (like V , ∆ϑ and W ), the solution space

may represented by a non-smooth manifold and the appearance of the stable closed

orbits is interpreted as a ”shift of stability” from the original stationary solution to the

periodic one [24], [16]. This approach reveals the potential of capturing from a general

point of view the crack behaviour, according to the imposed external conditions. The

mathematical concept is explained and illustrated with an example at the section 4.4.

From the numerical point of view, a few references are available in the literature.

Among them, we cite a couple of attempts to simulate the experimental results de-

scribed in the preceding paragraph. By adopting a peridynamic theory Kilic et al. in

[22] reproduce complex crack patterns by mean of a damage model. The technique

has the limit of not allowing the use of a generic Poisson coefficient (it is constraint

to 0.25 by the general assumptions of the theory). Furthermore, like other damage

models, it is able to describe a local reduction of the stiffness of the material but not

to explicitly model the fracture as a discontinuity produced in the solid. In order to

obtain meaningful results the authors needed to assign a slope to initial notch and

apply values of ∆ϑ and in particular of V much higher than the experimental ones.

In [11] Ferney et al. document some examples of oscillating and branched cracks

modelled by cohesive finite element. The simulation does not follow the experimental

condition: the expedient of increasing the descent speed (of six orders of magnitude)

changes the process from quasi-static to dynamic.

1.3 Analysis and modeling of the problem

1.3.1 Dimensional analysis

The number of numerical analysis that we are able to perform is limited by the com-

putational cost of each of them; at the same time we want to be able to compare the

numerical results with the large number of experimental data available in literature.

22

A possible way is to perform a dimensional analysis and use dimensionless parameters

to characterize the system and represent the outcomes.

The geometry of the problem is described by the following dimensionless param-

eters (being D the thickness of the plate):

i)D

L(D tends to be very small compared to its length).

ii)W

L(the ratio between the two relevant dimensions of the plate);

iii)H

L(the spatial gap between hot and cold regions respect to the total sample

length).

The heat problem is essentially governed by the Biot and the Peclet numbers

iv) Bi =hD/2

k(which describes the conduction problems that involve surface con-

vection effects),

v) Pe =HV

κ(which the ratio of the rate of advection of a physical quantity by

the flow to the rate of diffusion of the same quantity driven by an appropriate

gradient. In the context of the transport of heat, the Peclet number is equivalent

to the product of the Reynolds number and the Prandtl number).

In the definitions h is the convection heat transfer coefficient, k is the thermal con-

ductivity and κ the thermal diffusion coefficient.

This two values are very relevant in understanding the interaction between therma

and mechanical problem. More detailed considerations will follow in the next two

sections.

To summarize the information about the fracture traveling along the specimen,

we will make use of the following dimensionless quantities:

vi) tnm =V t

L(the dimensionless time);

vii) arel =a

L(the fracture length divided by the plate length);

viii) arel =a

V(the fracture propagation velocity compared to the immersion one).

23

The quantities concerning the thermo-mechanical problem may be classified as follow

ix) ν (the Poisson coefficient);

x) α∆ϑ (which characterize the thermal volumetric deformation);

xi)V E

h∆ϑ;

xii)V Gc

Wh∆ϑ.

The last two parameter may be combined to formulate quantities that are more

meaningful, for capturing and communicating the results of our numerical studies.

A natural choice for one of them is represented by the ratio between an increment

of the elastic energy and the fracture energy necessary for a straight extension of the

crack surface. We may compute the variation of the strain energy per unit of plate

thickness

δEstrain = Eα2∆ϑ2W (1.7)

and compare it with the amount of energy released for propagating the fracture,

obtaining the dimensionless parameter τ

τ =E(α∆ϑ)2W

Gc

. (1.8)

The temperature gradient and the plate width may be chosen in order to keep the

same value of τ : in this way experimental as well as numerical results carried out on

a range of descent speed values are equivalent to a set of tests, conducted varying

the sample width. In our application we assume constant material properties: Young

modulus E = 70 GPa, the coefficient of linear expansion α = 9 ·10−6K−1, the thermal

diffusion coefficient κ = 0.47mm2s−1. In the literature ([32], [23], . . . ) the value of

the fracture energy has been analyzed as function of the temperature at the crack

tip and of the driving velocity. It is beyond the purpose of this work to consider the

dependence of the fracture energy on other parameters, so we will consider a constant

average value.

24

Figure 1-8: Geometry of the experimental setup.

The rate of the energy released by the propagating fracture may be compared to

the convective flux coming form the plate surfaces immersed into the water. The

definition of the relative dimensionless parameter may be the following

ζ =Gca

hW∆ϑ. (1.9)

The geometry of the palate considered in the numerical computations is described

by the following constant quantities (see Fig.1-8)

L = 60 mm (plate length);

W = 20 mm (plate width);

H = 20 mm (distance between the water bath and the heated region of the plate).

25

1.3.2 Two-dimensional modelling of the heat problem

The adoption of a two-dimensional model may simplify the analysis and reduce the

computational cost of the simulations.

This approximation may be justified by verifying the Biot number. It provides a

measure of the temperature drop in the solid relative to the temperature difference

between the surface and the fluid [2] [20]. The resistance to conduction within the

solid is much lower than the resistance to convection across the fluid boundary layer

if

Bi 1. (1.10)

In this case it is reasonable to assume a uniform temperature distribution within the

solid at any time during a transient process. The quantity to be verified is

Bi =h

k, (1.11)

where h is the convection heat transfer coefficient, k is the thermal conductivity and Lc

is the characteristic length of the heat problem involving conduction and convection.

It corresponds to one half of the thickness of the plate for the specific geometry of

our problem (plane vertical wall). The data for the quenched plate are:

h ∼ 101 W/m2K (1.12)

k = 0.65 W/m K (1.13)

Lc = 0.5 · 10−4 m (1.14)

Bi ∼ 10−3 < 0.1 (1.15)

The order of magnitude of the Biot number allows the assumption of a constant

temperature field within the thickness of the plate.

26

1.3.3 Steady thermal profile

In the problem at the hand, the thermal profile along the plate is related to the

temperature difference ∆ϑ and the spatial gap H between the two reservoir of the

plate: the first is kept at a constant higher temperature Th, the second is immersed

into the the water at temperature Tc. A steady state can be characterized by two

constant temperature fields connected by a linear profile along the separation gap.

The driving velocity V localizes the thermal gradient in proximity the cold bath over

an area measured by the thermal diffusion length. The steady thermal regime may

be verified comparing the spatial gap with the thermal diffusion length, obtaining a

condition over the immersion velocity, which needs to be low enough to allow the

assumption of quasi-static condition. We can verify weather the Peclet number is

much lower than the unity or equivalently

H κ

V⇒ V κ

H(1.16)

V 0.47mms−1

10 mm= 0.047 mms−1 (1.17)

We limited the range of velocities of our simulations, adopting 0.01 mms−1 as max-

imum value in order to reasonably assume a steady temperature profile and reduce the

computational cost of the analysis. In the experiments of Yuse and Sano the velocity

reaches the value of 50 mms−1 and manifest crack patterns more and more regular

with the increasing of the descent velocity. This apparent antithetical behaviour is

due to the increasing complexity of the thermo-mechanical problem, according with

two main possible conditions:

• κ

h< V <

κ

Dthe heat conduction is no longer in steady regime and the temper-

ature gradient localizes near the cold bath over the extent of dth

• V >κ

Dthe thermal diffusion length becomes smaller than the thickness of the

plate: the temperature is no longer constant across the thickness, the fracture

process becomes three-dimensional and the resulting crack surface is rough.

The localization of the temperature gradient in the proximity the cold bath may

27

be obtained increasing the driving velocity as well as the distance between the two

reservoirs. Aiming to approximate a steady thermal regime, it is possible to perform

slower plate immersions or to reduce the area of the diffusion, tuning the H parameter

As already observed by [34].

The assumption of a steady thermal regime with perfect thermal baths and a

constant temperature gradient in the H region produces a discontinuous temperature

gradient field, leading to an overestimate of the stress field concentration at the crack

tip interface. The consequence could be a non precise prediction of the crack tip

position, influenced anyway by numerical constraints coming from the discretization

of the domain.

1.3.4 Organization of the contents

This research has been developed with the intention to solve the thermo-mechanical

problem through a finite element code, in a two-dimensional approximation of the

spatial set and imposing a steady temperature profile. The fracture propagation in

brittle materials has been modeled explicitly with cohesive finite elements in order

to predict the crack pattern on the basis of the stress concentration at the crack

tip. The instability of the straight crack propagation is derived as an outcome of the

numerical analysis, not requiring the formulation of an analytical kinking criterion.

This allowed us to avoid the necessity of severe simplification of the problem, such

as the hypothesis of infinite plate width that remove the possibility to capture the

dependency of the crack path on this geometric parameter.

The contents are organized according to the following subject subdivision. The

second chapter describes the general formulation of the thermodynamic problem. The

third chapter deals with the spatial discretization and the numerical tools adopted to

solve the thermo-mechanical problem and to track the evolution of the crack path.

The fourth chapter collects the most relevant results of numerical simulation of the de-

scribed experiments. Criticality and open issues arising from the analysis are pointed

out. The fifth chapter introduce current developments of this research and possible

future extension in the problem treatment.

28

Chapter 2

Non-linear Solid Mechanics

2.1 Thermodynamics of non-linear elastic materi-

als

2.1.1 Introduction

The continuum mechanics (Truesdell and Noll [40], Gurtin et al. [17], Holzapfel [18]) is

based on physical entities with a mathematical description. It is essential to consider

the concepts of body, regarded as a smooth manifold of material points; configuration

of a body, namely a mapping of the body in a three-dimensional Euclidean space;

force system acting on a body, defined by a vector-valued function. As constitutive

assumptions, force systems may include body forces with a mass density and contact

forces, based on a surface density and local properties of the configuration at the

point. The thermodynamics of continua includes the concepts of temperature, specific

internal energy, specific entropy, heat flux, and heat supply (due to radiation). As

general principles of mechanics we take into account the objectivity and the law

of balance of linear momentum; we add the fist law of thermodynamics (the law of

balance of energy) and the second law of thermodynamics in the form of the Clausius-

Duhem inequality. The constitutive assumptions for the material and the purposes

of this study are the following:

29

1. the specific internal energy is a function of the deformation gradient and of the

specific entropy;

2. the temperature is related to the deformation gradient and the internal entropy;

3. the stress tensor describes an elastic behavior based on the deformation and the

internal entropy, disregarding plastic and viscous effects;

4. the heat flux only depends on the deformation, the specific entropy and the

spatial gradient of the temperature.

Necessary and sufficient restrictions are derived imposing the Clausius-Duhem in-

equality valid for all the admissible processes.

2.1.2 Thermodynamic processes

A thermodynamic smooth process is a time-dependent set of configurations, force

systems, temperature, internal energy, entropy, heat supply and heat flux compatible

with the principles of mechanics and the law of conservation of energy. A thermody-

namic process is admissible when it is compatible with the constitutive assumptions

under consideration. Consider a body consisting of material points. It undergoes a

thermodynamic process described by eight function of the position X ∈ B ⊂ R2 of

the material point along the time t ∈ (0, T ) ∈ R+:

1. the spatial position x = χ(X, t), where χ is the motion of the body and it is

also called configuration mapping;

2. the body force B = B(X, t) per unit of mass;

3. the first Piola Kichhoff stress tensor P = P(X, t);

4. the internal energy density ε = ε(X, t);

5. the heat flux vector Q = Q(X, t),

6. the heat supply R = R(X, t) per unit of mass and unit of time

30

7. the specific entropy η = η(X, t)

8. the local temperature ϑ = ϑ(X, t) assumed to be always positive

When associated to a thermodynamic process the eight functions must satisfy the

four conservation laws applied to the whole body and to any sub-part Ω :

1. the law of balance of mass

D

Dt

∫Ω

ρ(X, t) dV = 0 (2.1)

2. the law of balance of linear momentum

∫Ω

ρ0χ dV =

∫Ω

ρ0B dV +

∫∂Ω

PN dS (2.2)

3. the law of balance of angular momentum

∫Ω

χ× ρ0χ dV =

∫Ω

χ× ρ0B dV +

∫∂Ω

χ×PN dS (2.3)

4. the law of balance of energy

D

Dt

∫Ω

(1

2ρ0χ

2 + ε

)dV =

∫Ω

(Bχ +R) dV +

∫∂Ω

(Pχ−Q) N dS (2.4)

In (2.2), (2.3) and (2.4), dV represents the element of volume of the body in the

reference configuration, ∂B the boundary of B , dS the element of surface in the

configuration at the reference time, the vector N the normal to the surface dS, the

superimposed dots denote the time-derivatives. In the preceding equations we made

use of ρ0 and ρ to denote the mass per unit of volume computed in the material and

spatial configuration respectively.

In the notation we adopt Greek and capital Latin symbols to refer to the un-

deformed configuration of the body, the so called material or Lagrangian reference.

Lower Latin case refers to the spatial or deformed configuration, with the exception

of cases when clarity suggests a different choice.

31

Under suitable compatibility assumptions (the presence of discontinuities due frac-

ture is taken into account and discussed in the next sections) the three balance integral

equations (2.2), (2.3) and (2.4) may be formulated locally in differential form:

Div P + ρ0B = ρ0χ (2.5)

PFT = FPT (2.6)

ε = P : F−Div Q + ρ0R (2.7)

where F denotes the time derivative of the deformation gradient F, defined as follow:

F(X, t) =∂χ(X, t)

∂X= Grad x(X, t) ∈ GL+(3,R) (2.8)

where GL+(3,R) is the Lie group of invertible and orientation preserving linear trans-

formations in R3. It is sufficient to prescribe the six functions χ,P, ε,Q, η, ϑ to define

a thermodynamic process and to uniquely determine B, R.

2.1.3 Constitutive equations

The constitutive assumptions define specific material properties for the body, accord-

ing with the application under consideration. An elastic material with heat conduc-

tion is defined by four response functions:

ε = ε(F, η) (2.9)

ϑ = ϑ(F, η) (2.10)

P = P(F, η) (2.11)

Q = Q(F, η,Gradϑ) (2.12)

Admissible processes are identified by prescribing a mapping χ(X, t) and an entropy

distribution η(X, t). The deformation gradient is computed from χ and ε, ϑ,P,Q

can be determined through constitutive laws. It follows that the value of B, R can be

chosen so that the balance equations hold. The principle of material frame indifference

32

(or objectivity) states that the admissibility of a process must not be affected by a

change of frame or observer. According to this principle the constitutive assumptions

must result to be indifferent to a change of frame, for example by defining the functions

in terms of the right Cauchy-Green tensor C = FTF

ε = ε(C, η) (2.13)

ϑ = ϑ(C, η) (2.14)

P = P(C, η) (2.15)

Q = Q(C, η,Gradϑ) (2.16)

(2.17)

2.1.4 Entropy inequality principle

The second law of thermodynamics postulates that the total production of entropy

is always non negative. Regarding Q/ϑ as a flux of entropy and ρ0R/ϑ as a supply

of entropy, the second law of thermodynamics takes the form of the Clausius-Duhem

inequality:

Γ(t) =D

Dt

∫Ω

η dV +

∫∂Ω

Q

ϑ·N dS −

∫Ω

R

ϑdV ≥ 0 (2.18)

The specific production of entropy is represented by the differential form:

η − ρ0R

ϑ+

1

ϑDiv Q− 1

ϑ2Q Gradϑ ≥ 0 (2.19)

An alternative local form is derived by substituting the energy balance into the equa-

tion

P : F− ε+ ϑη − 1

ϑQ Gradϑ ≥ 0 (2.20)

where the last term represents the entropy production due to heat conduction. The

chain rule applied to the internal energy rate reads

ε =∂ε

∂F: F +

∂ε

∂η: η (2.21)

33

Let us consider a precess defined by homogeneus deformations and entropy ditribu-

tion. Thus F, η may be chosen to be arbitrary functions of time and Grad Q = 0.

The Clausius-Duhem inequality becomes:

[ϑ(F, η)− ∂ε

∂η(F, η)

]η +

[P(F, η)− ∂ε

∂F

]: F ≥ 0 (2.22)

and in order to hold for any homogeneus process we must have

ϑ = ϑ(F, η) =∂ε

∂η(2.23)

P = P(F, η) =∂ε

∂F. (2.24)

The temperature and stress relations (2.23), (2.24) and the heat conduction inequality

−Q Gradϑ ≥ 0 (2.25)

may be regarded as sufficient conditions for the validity of the Clausius-Duhem in-

equality, when applied to any admissible thermoelastic processes. Inequality (2.25)

states that the propagation of the heat flux is in the direction of a negative temper-

ature gradient and it states the non-existence of a piezo-caloric effect, namely the

absence of heat-conduction in case of a constant temperature field. The temperature

relation can be inverted whenever ε is a convex function of η for a fixed F, so that

eq. (2.23) can be solved for the entropy

η = η(F, ϑ) (2.26)

Introducing the Helmholtz free energy through the Legendre transformation

ψ(F, ϑ) = infηε(F, η)− ϑη (2.27)

34

we derive an alternative form for the equations (2.23), (2.24)

η = η(F, ϑ) = −∂ψ∂ϑ

(2.28)

P = P(F, η) =∂ψ

∂F. (2.29)

The constitutive equations (2.24), (2.23) can be formulated taking into account the

material frame indifference, so that the second Piola-Kirchhoff stress tensor and the

entropy are defined through the free energy in the form:

S(C, ϑ) = 2ψ(C, ϑ)

∂C(2.30)

η(C, ϑ) = −ψ(C, ϑ)

∂ϑ(2.31)

depending only on C and ϑ.

2.1.5 Thermoelastic field equations

The energy balance can be formulated in terms of entropy in order to obtain a more

suitable equation for the statement of the thermoelastic problem. By introducing the

Gibbs relations

ψ =1

2S : C− ηϑ (2.32)

ε =1

2S : C + ϑη (2.33)

derived from the thermodynamic restrictions considered above, it is possible to sub-

stitute the energy balance with the entropy balance

η = −1

ϑDiv Q + ρ0

R

ϑ(2.34)

The expression of the entropy production for a thermo-mechanical process becomes

Γ = −1

ϑQ(C, ϑ,Gradϑ) Gradϑ (2.35)

35

The stress-temperature modulus

M(C, ϑ) =∂S(C, ϑ)

∂ϑ(2.36)

is defined as the tensor that measures the change in stress due to a change in tem-

perature (at fixed strain) and the heat capacity (at fixed strain) Cv, defined as the

specific heat per unit of mass

ρ0Cv(C, ϑ) =∂ε(C, ϑ)

∂ϑ= −ϑ∂

2ψ(C, ϑ)

∂ϑ2(2.37)

Differentiating the entropy with respect to the time we get

ϑη = ϑ∂η(C, ϑ)

∂C: C + ϑ

∂η(C, ϑ)

∂ϑϑ (2.38)

= −1

2ϑM(C, ϑ) : C + c(C, ϑ)ϑ (2.39)

Accounting for the frame indifference and the respect of the thermodynamic princi-

ples, the evolution equation for the temperature can be written as follow

ρ0Cv(C, ϑ)ϑ = −Div Q(C, ϑ,Gradϑ) +1

2ϑM(C, ϑ) : C + ρ0R (2.40)

In the classical formulation of the heat conduction problem, the heat capacity is

considered constant and the heat flux is expressed by the Fourier law, often applied to

an isotropic material. The strain rate driven by the immersion velocity V considered

in the present study supports the hypothesis of a negligible value for the second term

on the right hand side of (2.40). Summarizing, the linear momentum balance and the

energy balance (in terms of entropy balance) are

Div P + ρ0B = ρ0χ (2.41)

ρ0Cvϑ = k4ϑ+ ρ0R (2.42)

where the simbol 4 represents the laplacian of the absolute temperature ϑ.

36

The set constitutive equations reads:

ψ = ψ(C, ϑ) (2.43)

P(C, ϑ) = 2Fψ(C, ϑ)

∂C(2.44)

η(C, ϑ) = −ψ(C, ϑ)

∂ϑ(2.45)

Q(Gradϑ) = −K Gradϑ, (2.46)

where the thermal conductivity tensor K(X) is assumed to be positive definite and

takes the form

K(X) = k(X)I (2.47)

for isotropic materials.

The simplification adopted for the present study lead to uncoupled governing

equations: the problem can be solved computing separately the temperature field

and using the temperature as a data for the solution of the mechanical response.

2.1.6 State functions for thermo-elastic materials

In the kinematics it is possible to adopt an operative point of view and decouple

the volumetric inelastic thermal deformation gradient Fa and the elastic deformation

Fe, which represents the local stretch and distortion of the material lattice. When

the external mechanical load is null, the elastic deformation is caused by the incom-

patibility of the thermal deformation field that prevents the full relaxation of the

inelastic deformation due to the variation of the temperature in time. According to

standard approach documented in the literature, we assume locally a multiplicative

decomposition of the deformation gradient, as proposed by Kroner (1960) and Lee

(1969)

F = FeFa (2.48)

37

Additionally we assume a form for the free energy, as presented by Yang and Ortiz

[42], Stainier and Ortiz [38]

ψ(Fe, ϑ) = ψe(Fe) + ρ0Cvϑ

(1− log ϑ

ϑ0

)(2.49)

where ψe is the elastic strain energy density, ϑ0 is a reference temperature. The strain

energy density ψe depends solely on the elastic elastic part of the deformation tensor;

material-frame indifference requires the dependence on the elastic right Cauchy-Green

tensor

Ce = FeTFe = Fa−TCFa−1 (2.50)

2.1.7 Constitutive law: Isotropic Hyperelastic Model

The actual choice for the strain energy density is a neoHookean potential, extended

to the compressible range:

ψe(Fe) =1

2λ log2 Je − µ log Je +

1

2µ(Ie1 − 3)

The choice of a multiplicative decomposition leads to the particular formulation of

the total stress in the form

P =∂ψe

∂F=∂ψe

∂Fe

∂Fe

∂F(2.51)

where:∂Fe

∂F= Fa−1. (2.52)

The elastic first Piola-Kirchhoff stress tensor reads:

Pe =∂ψe

∂Fe =∂ψ

∂Je∂Je

∂Fe +∂ψ

∂Ie1

∂Ie1∂Fe = (λ log Je − µ) Fe−T + µFe (2.53)

38

The spatial elasticity tensor is:

Ce =∂2ψ

∂Fe∂Fe =∂Pe

∂Fe =∂Pe

∂Je∂Je

∂Fe +∂Pe

∂Fe−T∂Fe−T

∂Fe +∂Pe

∂Fe (2.54)

= λFe−T ⊗ Fe−T − (λ log Je − µ) Fe−1 Fe−1 + µI (2.55)

The Piola-Kirchhoff stress tensor is:

P =∂ψe

∂Fe

∂Fe

∂F=[(λ log Je − µ)Fe−T + µFe

]Fa−1. (2.56)

The Cauchy stress σ is recovered as:

σ = J−1PFT = J−1[(λ log Je − µ)Fe−T + µFeT

]Fa−1FaTFeT . (2.57)

Finally, the referential stiffness tensor is:

C =∂2ψ

∂F∂F=∂P

∂F=

∂P

∂Fe

∂Fe

∂F=

(∂Fe

∂F

)T∂Pe

∂Fe

∂Fe

∂F= Fa−TCeFa−1. (2.58)

The second Piola-Kirchhoff stress tensor is recovered as

S = F−1P = Fa−1Fe−1[(λ log Je − µ)Fe−T + µFe

]Fa−1 (2.59)

= Fa−1[(λ log Je − µ)Fe−1Fe−T + µI

]Fa−1 (2.60)

= Fa−1[(λ log Je − µ)Ce−1 + µI

]Fa−1. (2.61)

A simple choice for the thermal deformation gradient is to assume a fully volumetric

expansion, therefore we write:

Fa = (1 + αδϑ) I (2.62)

Fa−1 =1

(1 + αδϑ)I (2.63)

39

the second Piola-Kirchhoff stress tensor may be computed as follow

S =1

(1 + αδϑ)2

[(λ log Je − µ)Ce−1 + µI

](2.64)

Recalling the definition of the stress-temperature modulus

M(C, ϑ) =∂S(C, ϑ)

∂ϑ=

−2α

(1 + αδϑ)

[(λ log Je − µ)Ce−1 + µI

](2.65)

we obtain an expression that may be use in the heat equation (2.42) in cases where

the strain rate becomes relevant for the problem, i.e. if the immersion velocity V

takes values with order of magnitudes 3− 4 times larger than the one we are focused

on.

2.2 Fracture

2.2.1 Irreversible Cohesive Fracture model in a Non-linear

kinematics framework.

A simple cohesive law allows to describe fracture nucleation ad propagation through

only two material parameters: the peak cohesive traction and the fracture energy

of the material. The cohesive theory of fracture since its first theorical formulation

due to Dugdale [10] Barenblatt [1], Rice and others regards the fracture as a gradual

phenomenon in which separation takes place across an ”extended” crack tip (cohesive

zone) and it is resisted by cohesive tractions t. This approach can be endowed into

the material independently of its peculiarity like the constitutive behaviour of the

bulk or the characteristic size of the process zone. The process zone defines a region

around the crack tip in which inelastic deformation takes place and where the fracture

energy is dissipated through a progressive de-bonding that leads to the formation

of new surfaces. Traditionally, in finite element analysis, cohesive approaches have

been embedded in specific cohesive finite elements [4] [28]. Surface-like elements are

compatible with the general bulk discretizations of the solid.

40

Figure 2-1: Cohesive surface traversing a 3D body

We assume that the volume Ω of the body B is traversed by a cohesive surface

Scoh which partitions the body into two subbodies Ω±, lying on the the two sides of

Scoh. The deformation power, i.e., the power committed by the system in the elastic

and inelastic deformations, is then represented by the part of the mechanical power

expended on the solid by external (volume and surface) forces non applied in rising

its kinetic energy:

ψ = ψext − K =∑±

∫Ω±

ρ(B− χ) · χ dV +∑±

∫∂Ω±

T · χ dS (2.66)

The kinetic energy is given by the sum of the quantities computed over the two

sub-bodies

K =∑±

∫Ω±

1

2ρ | χ |2 dV (2.67)

The balance of tractions on the external surface of the sub-bodies and on the cohesive

interface:

PN = T on ∂Ω± (2.68)

JPNK = JtK = 0 on ∂S±coh (2.69)

41

leads to the strain energy for a cohesive solid of the form

∫Ω

P : F dV +

∫Scoh

t · JχK dS. (2.70)

From (2.70) the duality or work-conjugacy relations between stress and strain mea-

sures can be discerned by distinguish:

• over the bulk of the body, as in conventional solids, the first Piola-Kirchhoff

stress tensor P is work-conjugate to the deformation gradients F;

• over the cohesive surface, the tractions t are work-conjugate to the displacement

jumps or opening displacement δ = JχK.

The second integral in eq. (2.70) is defined over Scoh , which, given the presence of

two flanks, needs to be defined explicitly. We refer to the cohesive surface as the

”average” of the two interface surfaces Scoh+ and Scoh

−

χ =1

2(χ+ + χ−) (2.71)

and the mapping of the two flanks becomes:

χ± = χ± 1

2δ (2.72)

Pandolfi and Ortiz [28] postulate the existence of a free energy density per unit of

undeformed area over Scoh in the general form

φ = φ(δ, ϑ,q; Gradχ) (2.73)

as a function of the local temperature ϑ and of a collection of suitable internal vari-

ables q able to describe the inelastic process of decohesion. The surface deformation

described by Gradχ is included into the formulation to take into account possible

anisotropic behaviour of the material, since for isotropic materials a change of χ at

constant δ does not imply any decohesion or crack closure. By assuming that the

42

cohesive behavior derives from a free energy density φ, the cohesive law reads:

t =∂φ

∂δ(2.74)

Irreversibility requires the definition of an evolution law for the internal variables q,

governed by a kinetic relation of general form

q = q(δ, ϑ,q). (2.75)

For simplicity we disregard the dependencies on the temperature and on the deforma-

tion of stretching and shearing of the cohesive surface. It follows that the free energy

density must be of the simpler form

φ = φ(δn, δ ·Gradχ,q). (2.76)

Material isotropy allows to assume the response to sliding as independent of the

direction and to consider only the dependence on the the norm | δS |= δS of the

sliding component of the crack opening displacement. The energy density and surface

traction can be written as

φ = φ(δn, δS,q) (2.77)

t =∂φ

∂δn(δn, δS,q)n +

∂φ

∂δS(δn, δS,q)

δSδS. (2.78)

In [4] a further semplification was introduced by reducing the opening displacements

to a scalar valued effecttive one

δ =√β2δ2

S + δ2n (2.79)

which is adopted also in the expression of the fre energy

φ = φ(δ,q). (2.80)

43

Figure 2-2: Adopted cohesive law expressed in terms of an effective opening dis-placement δ and a traction t: loading-unloading rule from linearly decreasing loadingenvelop.

Under this condition the cohesive law assumes the expression

t =t

δ(β2δS + δnn) (2.81)

where t is the scalar effective traction derived from φ through the effective opening

displacement

t =∂φ

∂δ(δ,q) =

√β−2 | tS |2 +tn

2 (2.82)

The factor β defines the ratio between the shear and the normal critical tractions.

Upon closure the cohesive surfaces are subjected to the contact unilateral constraint

including friction. These surfaces interactions may be modeled as independent phe-

nomena, but the aim of the present work is simply to characterize the cohesive law

with an irreversibility property by defining a suitable unloading law. An appropriate

choice of the internal variable set q reduces to the maximum ever attained effective

opening displacement δmax. Loading is characterized by the conditions

δ = δmax ; δ ≥ 0 (2.83)

We adopt a cohesive law represented by a piecewise linear function, as depicted

in Fif.2-2. The sketch illustrates the rule of unloading to the origin, as introduced in

44

[4], i.e.,

t =Tmaxδmax

δ if δ < δmax ; δ < 0 (2.84)

45

46

Chapter 3

Numerical procedure

3.1 Heat conduction problem

Let us recall the equations of the energy balance (2.41), (2.42)

Div P + ρ0B = ρ0χ (3.1)

ρ0Cvϑ = k4ϑ+ ρ0R (3.2)

The energy balance equation represents the heat conduction law of a solid and it

may be recast as follow

ϑ = κ4ϑ+R

Cv(3.3)

Eq. (3.3) represents the diffusion parabolic equation of the transient heat problem.

Assuming no source of radiant heat into the body and we consider negligible the time

derivative of the temperature ϑ, adopting the elliptic Laplace equation of the steady

heat problem. This choice is supported by the analysis of the Biot number relative to

the particular set of experimental results which is the scope of the present research.

For more detail we refer to the section 1.3.3. The formulation of the boundary value

47

thermal problem [35], [30] may be written

−κ4ϑ = 0 in Ω (3.4)

ϑ = ϑ0 on Γcold (3.5)

ϑ = ϑ0 + ∆ϑ on Γhot (3.6)

The Dirichlet non-homogeneus boundary conditions (3.5) and (3.6) are represented

by the temperature we impose to the node belonging the cold subdomanin Γcold

(supposed to be at water temperature) or to the hot one Γhot (supposed to be kept at

the oven temperature). The solution of the problem is well known and it is represented

by a linear distribution of the temperature in the spatial gap between the cold and

hot reservoir where the temperature is imposed to be constant. This assumption has

been justified by analysing the conditions for thermal process in the section 1.3.3.

3.2 Weak form and discretization of the mechani-

cal problem

3.2.1 Weak Form

We restate the linear momentum balance (2.41) in index notation, imposing the

boundary conditions and removing the body force term, not relevant for this problem.

Div P = ρ0χ in B × (0, T )

χ = χ on ΓD × (0, T )

PN = T on ΓN × (0, T ) Neumann boundary

where ΓD and ΓN are on disjoint part of the boundary of B representing respec-

tively the Dirichlet and Neumann boundary. The non-linear mechanical problem is

defined on the domain B ⊂ R2 and over the time interval (0, T ) ∈ R+, with mixed

48

boundary conditions represented by displacements and tractions imposed ΓD and ΓN

and completed by the initial conditions

χ|t=0 = χ0

χ|t=0 = χ0

To obtain the equivalent variational form of problem that is amenable to discretiza-

tion, we introduce the power functional J : X 7→ R

J(χ) =

∫ΓN

T · χ dS︸︷︷︸W ext

− d

dt

∫B

1

2ρ0‖χ‖2 dV︸︷︷︸K

−∫B

ψ dV︸︷︷︸W int

(3.7)

with energetic aspects of the problem condensed in the following terms:

• W ext is external power expended on body.

• K is rate of change of kinetic energy of body

• W int is the power arising from the deformation of the body. It includes no form

of dissipation, to be consistent with the reversible modeling here in force.

In the next equations the arguments (X, t) of all fields are suppressed for the sake of

clarity.

Admissible trial deformation mappings belong to a non-linear space

X =χ ∈ [H1(B)]2 × C1((0, T ))

∣∣χ|ΓD= χ

(3.8)

due to the presence of inhomogeneous boundary conditions. Function space of ad-

missible spatial variations is

V =

v ∈ [H1(B)]2 × C0((0, T ))∣∣v|ΓD

= 0

(3.9)

The equivalent variational form of the momentum balance states that the first

variation of the power functional vanishes when tested against all admissible variations

49

i.e.

〈DJ(χ); η〉 = 0 ∀η ∈ V (3.10)

Taking the directional derivative of the energy functional we obtain

〈DJ(χ); η〉 =d

dεJ(χ + εη)|ε=0 (3.11)

=

∫ΓN

T · η dS −∫B

ρχ · η dV −∫B

P : Grad η dV (3.12)

Hence the weak form of the mechanical problem is find χ ∈ X such that

∫ΓN

T · η dS −∫B

P : Grad η dV =

∫B

ρ0χ · η dV (3.13)

for all η ∈ V .

3.2.2 Spatial Discretization

To be able to adopt numerical tecnique and computational implementation for solving

the mechanical problem, we perform a semi-discretization in space of the weak form

3.13 with the finite element method.

We form a shape-regular, pairwise disjoint triangulation T of B, comprising tri-

angles K. We choose standard discrete function spaces Xh ⊂ X and Vh ⊂ V to

interpolate field variables

Xh = χh ∈ X|χh|K = Pn ∀K ∈ Th (3.14)

Vh = v ∈ V |vh|K = Pn ∀K ∈ Th (3.15)

As in the standard Galerkin finite elements formulation, the function spaces Vh and

Xh coincide (apart from Dirichlet boundary). In all the simulation we adopt quadratic

polynomial interpolating functions, so we have n = 2.

The finite element approximation of the body motion is described as a linear

50

combination of the nodal values of the function

χh =∑b

N b(X)xb(t) η =∑a

Na(X)ηa(t) (3.16)

(3.17)

where Xb and ηa are the nodal coefficients of the basis functions Na, N b ,with indices

a, b each ranging over all the nodes in the mesh.

The time derivatives follow

χh =∑b

N bxb χh =∑b

N bxb (3.18)

likewise Grad η =∑

a GradNa ⊗ ηa.

Substituting the definitions into 3.13, we obtain a discreat version of the weak

form

∑a

ηai

[∫ΓN

TiNa dS −

∫B

(Ph)iJ Na,J dV −

(∑b

∫B

ρ0NaN b dV xbi

)]= 0i (3.19)

which must hold for all η ∈ Vh. The discrete Piola-Kirchoff stress is computed directly

from the discrete deformation mapping using

Ph = P(Fh), Fh = Gradχh =∑b

GradN b ⊗ xb

Eq. 3.19 places no constraint on the ηai , and hence the above holds if and only if

each term of the sum vanishes independently. We express this condition in a compact

matrix form

Faext − Fa

int = Mabxb (3.20)

with the following definitions

• external forces: F ai ext =

∫ΓNTiN

a dS

51

• internal forces: F ai int =

∫B

(Ph)iJ Na,J dV

• mass matrix: Mab =∫Bρ0N

aN b dV

and implying the summation over nodal index b. These calculations are carried out

on an element-by-element basis. According to the standard F.E. approach, the local

contribution is computed by restricting the support of the basis functions to the single

element and summed to the global arrays above at shared nodes.

The semi-discrete equations (3.20) describe a transient dynamic process, so it

needs to be discretized also in time. For the static problem we are dealing with

null acceleration. The necessity of including the term relative to the acceleration

is justified by the numerical strategy applied for solving the static problem. The

dynamic relaxation technique, indeed, regards the statical solution as the steady-

state response of the system, combined with the explicit Newmark algorithm to carry

the time integration.

Operatively, the mass matrix is not calculated using an approximation that ren-

ders the matrix diagonal. This simplification leads to large savings in computational

effort, reducing the solving system into a set of independent equations. In general

the substitution of the consistent mass matrix with the diagonal lumped one does

not affect the accuracy of the solution in a relevant way, while it helps in reducing

the computational time [19] It is particularly favorable in our case, since the specific

transient solutions represent an artificial expedient.

3.3 Dynamic Relaxation

The problem we are analyzing represents an example of a system in quasi-static

conditions undergoing instability phenomena. This means that the algorithm solving

the mechanical problem, has to be able to detect possible bifurcation points and to

follow the evolution of the unstable behaviour.

A suitable tool for this purpouse is the dynamic relaxation tecnique, which con-

sistes of determining the steady-state response to the transient dynamic analysis for

52

an autonomous system. In this case the transient part of the solution is of no interest,

the desired outcome is represented by the steady-state response of the system.

Regarding the whole problem a reduced loading time step is usually required to

ensure the capturing of particular configurations. The computational cost per time

step is minimal and is mostly associated with the evaluation of the internal force

vector Fint. The assembly and storage of a global stiffness matrix is not required

and this allows the treatment of complex material models. Being a critically damped

process, the dynamic relaxation algorithm is generally slow but reliable and it has the

potential for efficient implementation on a wide range of high-performance parallel

computer architectures.

The discretized equations of motion governing the dynamic response, for the nth

artificial time increment, may be written as follow:

Mxn + cMxn + Fint(xn) = Fn

ext (3.21)

where M is a mass matrix, c is the damping coefficient for mass-proportional damp-

ing, Fint is the internal force vector, and Fnext is a vector of external loads at the

current time step. The vectors xn, xn and xn represent the acceleration, velocity

and displacement vectors, respectively. A diagonal mass matrix obtained by mass

lumping, as well as a diagonal mass-proportional damping matrix, leads to a decou-

pled set of algebraic equations, where each solution component may be computed

independently.

In the present approach the solution of (3.3) is obtained by applying an explicit

time integration method with central-difference scheme (Newmark iterative proce-

dure).

˙xn+1/2 =1

∆t

(xn+1 − χn

)(3.22)

xn =1

∆t

(˙xn+1/2 − ˙xn−1/2

)(3.23)

where ∆t is the artificial time step. The implementation of dynamic relaxation en-

53

sures large stability limits and optional parallel implementation. The objective of the

adaptive dynamic relaxation algorithm for a static analysis is to reach the steady-state

solution of a critically damped pseudo-transient response. The mass and damping pa-

rameters do not represent the physical system, hence they are artificially defined in

order to produce the fastest convergence to the steady-state solution. The conver-

gence criterion is based on a relative error of the residual force:

ε =‖Fext − Fint‖‖Fext + Fint‖

≤ εtol (3.24)

3.3.1 Integration parameters

The dynamic relaxation integration parameters for static analysis consist of the diag-

onal mass matrix M, damping coefficient c and time step ∆t. They can be determined

on the basis of simple observations. The stability condition for explicit analysis im-

poses an upper limit for the time step driven by the minimum size of the finite

elements of the discretized body. The stable time step is estimated as the quotient

between the mesh size and the elastic longitudinal wave speed for the material:

∆t ≤ mine

(h

s

)e

(3.25)

A value of c which critically damps the system and implies a fast convergence to

the steady state is estimated by the following ratio, as documented by Oakley and

Knight in [27]:

c ' 2ω0ωm√ω2

0ω2m

(3.26)

being ω0 and ωm the minimum and maximum frequency of the discretized system.

The value of the minimum natural frequency of a mechanical system is usually much

smaller of the maximum one. This lead to the following approximated expression:

c ' 2ω0 (3.27)

54

An estimate of the minimum eigenvalue ω20 is represented by the stiffness-mass Rayleigh

quotient

ω20 '

vTKv

vTMv(3.28)

where v is a generic weighting vector and K is is the stiffness matrix (for linear

problems). The following diagonal approximation of the tangent stiffness matrix can

be assumed for non linear problems:

Kii =Fint i(x

n+1)− Fint i(xn)

xn+1 − xn. (3.29)

In particular, assuming as weighting vector the difference

v = xn+1 − xn (3.30)

the Rayleigh quotient becomes

ω20 = max

((xn+1 − xn)T(Fn+1

int − Fnint)

(xn+1 − xn)T ·M(xn+1 − xn), 0

)(3.31)

For problems which exhibit structural instabilities the stiffness matrix may loose

positive definiteness. When this occurs, the lowest eigenvalue may become negative

and the damping coefficient c is set equal to zero.

In non-linear problems the integration parameters must be updated in an adaptive

manner in order to maintain stable and efficient convergence. The version of Dynamic

Relaxation with parameter updates is called [41] Adaptive Dynamic Relaxation.

3.4 Fracture

The propagation of the fracture is explicitly implemented in the finite element model,

so that cracks can branch, coalesce and eventually produce fragments. The creation

of a new fracture surface is accomplished by allowing initially coherent inter-elements

55

(a) Geometry of cohesive elementcomposed by the surfaces S− S+

(b) Triangular cohesive element with12 node

Figure 3-1: Geometry of tringular cohesive element inserted between two 10-nodetetrahedral elements

boundaries to open according to a cohesive law which models a gradual loss of strength

with increasing flanks separation. The choice of the cohesive law defines the work of

separation, or fracture energy, required for creating a new free surface.

3.4.1 Finite element implementation of fracture

Rather than implementing the cohesive law as a mixed boundary condition we di-

rectly embed the cohesive law into surface-like finite elements. The cohesive elements

consists of two surface elements S− S+ (figure 3-1a) which concide in space in the

reference configuration of the solid. Each of the elements has n nodes and the total

number of nodes is therefore 2n. The figure 3-1b represents in particolar the ge-

ometry of triangular cohesive elements compatible with three-dimensional tetraedral

elements. We denote by Na(s1, s2), a = 1, . . . , n the standard shape functions of the

cohesive elements, being (s1, s2) the natural coordinates of each surface elements in

some convenient standard configuration. The calculation require a continuous track-

ing of the tangential and normal direction. Since S− S+ may diverge by a finite

distance, we uniquely define the normal n on the middle surface S, parametrically

56

Figure 3-2: (a) Standard element configuration and natural co-ordinate system; (b)deformed middle surface S and corresponding curvilinear coordinate system.

defined as

x(s) =n∑a=1

xaNa(s) (3.32)

where

xa =1

2

(xa+ + xa−

)(3.33)

and xa±, a = 1, . . . , n are the coordinates of the nodes of the cohesive elements in

their deformed configuration. The tangent basis vectors of the curvilinear coordinate

system defined by (s1, s2) for the middle surface correspond to

a,α(s) =n∑a=1

xaNa,α(s) α = 1, 2 (3.34)

The unit normal which points from S− to S+ becomes

n =a1 × a2

| a1 × a2 |(3.35)

In the spatial configuration we compute also the opening displacement vector (see

57

Figure 3-3: Opening displacement

figure 3-3)

δ(s) =n∑a=1

JxaKNa(s) (3.36)

where

JxaK = xa+ − xa− (3.37)

shows as δ is invariant upon sumperimposed rigid body motions. The cohesive trac-

tion vector compatible with the adopted model may be defined per unit of undeformed

area as follow

t = t(δ,n) =t

δ

[β2δ +

(1− β2

)(δ · n) n

](3.38)

The dependence of t on the normal needs to be take into account into the finite el-

ement implementation. The nodal forces arised from the tractions are computed as

integral extend over the undeformed surface of the element in its reference configura-

tion as follow

F ai± = ∓

∫Scoh

tiNadS (3.39)

There is no need to compute the stiffness matrix relative to fracture contribution,

the algorithm adopted for the mechanical problem does not require the solution of a

58

Figure 3-4: Flow chart of the numerical procedure for the solution of the thermo-mechanical problem and the detection of fracture propagation.

linear system of equations.

3.5 Synthesis of the numerical strategy

The numerical procedure implemented in the finite element code consists of a few

logical operations repeated at each time step. As first, evolving boundary conditions,

represented by nodal temperature, are applied according to the steady thermal profile

already described and motivated. The consequent inelastic volumetric expansions are

computed and imposed as input for the mechanical problem. Through the dynamic

relaxation technique the solution to the mechanical problem is obtained and in par-

ticular the stress field within the plate becomes know. The failure test is performed

at each element boundary interface by comparing the traction forces applied with a

critical value. If the test is positive a new cohesive element is inserted, the continuity

condition between the elements is removed and the topology is updated. Within the

same fracture iteration, a mechanical analysis follows the opening of a new facet in

order to re-compute the value of the deformation and stress tensors. At each time

step the failure test is repeated as long as its outcome is positive and the fracture can

59

be extended. If the test fails, no crack propagation needs to be accounted and the

procedure can move on to the next time step.

60

Chapter 4

Numerical results

4.1 Numerical test setup.

In order to simulate the instability of the direction of crack propagation, we have

developed a numerical code. Assumptions on the governing equations of our model

have been discussed in the previous sections. We proceed explaining some of the

details of the implementation and presenting the results so far produced.

The mechanical problems is discretized by a two-dimensional finite element model

with quadratic triangular elements. The heat conduction process is simulated by

imposing evolving temperature conditions, represented by an assigned thermal profile:

i) the water temperature ϑ0 imposed on the node gradually immersed into the bath;

ii) a linear increasing temperature (from ϑ0 to ϑ0 + ∆ϑ over the length H) in the

area between the cold and hot reservoir;

iii) the initial plate temperature ϑ0 + ∆ϑ associated to the node of the top part of

the sample.

This Dirichlet type boundary conditions change in time according to the descent

velocity characterizing the experiment.

The thermal strains, computed on the basis of the local temperature, are assumed

as datum of the mechanical problem. The stress state induced by inhomogeneous

61

thermal expansion develops in correspondence of the non-negligible second spatial

derivative of the temperature field and it is perturbed by the presence of the growing

crack.

An isotropic, non linear, hyperelastic model is adopted for the bulk material.

Dissipative behaviours only manifest in the process zone ahead of the crack tip and

it is accounted by the cohesive law. The cohesive model is based on the definition of

an effective fracture opening displacement δ, work conjugate to an effective fracture

traction t. The two scalar quantities are defined based on the corresponding vectors,

assigning different weights to the normal and to the tangential components. The

crack is allowed to propagate, from the initial notch tip, only between the interfaces

of the original elements, when the effective traction reaches a critical value defined

by the chose fracture criterion

t =√β−2 | tS |2 +tn

2 ≤ tcr. (4.1)

Fracture is then simulated via the local introduction of 3-node cohesive elements in

between the original elements, through an auto-adaptive re-meshing procedure. The

cohesive law governs the subsequent effective opening process along the new surfaces.

4.2 Qualitative discussion on the crack patterns

resulting from the simulations.

The main set of numerical experiments has been performed by keeping constant ma-

terial and geometry of the sample and varying the temperature gap ∆ϑ and the

immersion speed V , in order to investigate the dependance of the results on these two

parameters.

The Fig.4-1 summarizes the global fracture behaviour according to three classes

of patterns:

i) straight crack (symbol in Fig.4-1: +);

62

10−4

10−3

10−2

100

150

200

250

300

350

400

V [mm/s]

∆θ[K

]

κ/H ↑

StraightOscillatoryBranched

Figure 4-1: Phase Field of the crack morphology, depending on the parameters: Vand ∆ϑ. The value κ/H represents the limit of validity of the steady thermal stateassumption.

ii) oscillating crack (symbol in Fig.4-1: o);

iii) branched crack (symbol in Fig.4-1: x);

the combination of two symbols accounts for mixed patterns.

The cold reservoir is at the constant temperature 20 C, the temperature of the

hot region varies, according to ∆ϑ, assuming alternatively the values: 220, 270, 320,

370. Nine different immersion velocity over the range 0.0001mms−1 - 0.01mms−1 have

been tested.

We collect here the crack patterns relative to each simulated case, in order to

extract some considerations:

i) The finite element discretization of the problem, the non-linear kinematics de-

scription of the system and the cohesive elements regulating the failure at the

element interfaces have been able to capture oscillating and branched patterns.

ii) The computational cost of a large number of analysis has imposed a limit on

the adopted mesh size; despite of that the crack grows trying to respect the

symmetry respect to the vertical central axis, while following non-trivial paths.

63

(a)0.0001

(b)0.00025

(c)0.0005

(d)0.00075

(e)0.001

(f)0.0025

(g)0.005

(h)0.0075

(i)0.01

Figure 4-2: Fracture patterns resulting from ∆ϑ = 350 C. In correspondence of eachfigure the caption expresses the velocity in mms−1.

(a)0.0001

(b)0.00025

(c)0.0005

(d)0.00075

(e)0.001

(f)0.0025

(g)0.005

(h)0.0075

(i)0.01


iii) In Fig.4-2 the fracture patters are characterized by higher instability, most of

them showing an initial oscillating path. Later the fracture splits into two

branches, which propagate in almost straight and symmetrical directions. This

results correspond to the expectations: for ∆ϑ = 350 C the plate undergoes

more severe temperature curvature, so higher stresses arise from incompatible

deformations as well as a larger quantity of elastic energy is stored.

iv) Confirming experimental observation, for a fixed velocity, the wave length in the

oscillation increases with the temperature gap. In Fig.4-5 the waves are affected

by the mesh size, they tend to be small but over the limit imposed by the domain

64

(a)0.0001

(b)0.00025

(c)0.0005

(d)0.00075

(e)0.001

(f)0.0025

(g)0.005

(h)0.0075

(i)0.01


(a)0.0001

(b)0.00025

(c)0.0005

(d)0.00075

(e)0.001

(f)0.0025

(g)0.005

(h)0.0075

(i)0.01


(a)0.0001

(b)0.0005


65

discretization. In Fig.4-4 and above all Fig.4-3 the wave-length becomes larger

and larger. In the latter the mesh size still affects the path-shape, but with

minor relevance on the amplitude of the oscillation.

v) The incomplete set of simulations performed with ∆ϑ = 150 C are mostly char-

acterized by straight crack propagation, revealing important information about

the localization of the stability threshold. The identification of limit of tran-

sition from straight to oscillating patterns is a topic of high interest for the

prosecution of the present work.

4.2.1 Fracture propagation conditioned by mesh dependency.

For a fixed ∆ϑ, some of the obtained crack patterns show no changes by increasing

the velocity within a certain range. This effect is due to the restriction imposed by

the adopted discretization on the available fracture paths: the solution space is no

rich enough to allow alternative propagations in presence of small variations of the

loading conditions. For investigating this issue we performed the same analysis of

Fig.4-4 but using a mesh size one half of the previous one. The results collected in

Fig.4-7 clearly show how, enriching the solution space, the fracture path tends to

converge to the physical one, as well as the quantity of energy dissipated.

4.2.2 Crack morphology dependence on the immersion ve-

locity.

For a constant temperature jump ∆ϑ we analyze the evolution of the fracture pat-

terns with the descent velocity. For velocities lower then 10−1mms−1 we observe a

constance in the crack morphology, revealing a minor dependence of the propagation

pattern on V . When the velocity approaches the value of 10−1mms−1, or equiva-

lently when the Peclet number tends to 1, the trend changes toward an increasing

regularity in the crack path with V . We may interpret this results by comparing

them with the data available in literature. In [43], [44], (see Fig.1-4) we may focus

on the threshold lines between different crack morphologies. They are characterized

66

(a)0.0001mms−1

(b)0.00025mms−1

(c)0.0005mms−1

(d) 0.001mms−1 (e) 0.005mms−1

Figure 4-7: Fracture patterns resulting from ∆ϑ = 250 C

by a fast increasing of regularity, followed by a milder trend to irregularity. The

range of our velocities and the value of ∆ϑ chosen in our experiments collocate our

results in the left-top part of the graph, where in Fig.1-4 it is essentially recorded

oscillating and branched patterns, without a clear distinction between them. Close

to the transition threshold the same authors report the possibility of combination

and transition between the two patterns. In [32], [33], Fig.4-8 an interval of lower

immersion speed is considered 0.01 − 10 mms−1. The data are collected considering

the velocity versus the plate width. For V of orders 10−2 − 10−1 mms−1 a steady

thermal regime is established. The dependence on the velocity is negligible as long as

the thermal length dth is greater than the distance H between cold and hot baths. At

higher velocities the localization process results in decreasing the critical widths with

V , as long as dth remains larger than one half of the sample thickness. Under such

conditions the assumption of a steady thermal profile does not hold anymore and in

the numerical experiments it becomes necessary to perform a thermal analysis before

solving the mechanical problem, even if the system of thermo-mechanical equations

67

Figure 4-8: Experimental results of [34] relate the crack morphology to the platewidth and the driving velocity V . The treshold lines reflect the variation of the thermalfield with the V . The transition lines are controlled by (a) the distance H betweenthe reservoirs at low velocities; (b) the thermal diffusion lenght dth in the decreasingpart of the graph; (c) the thickness of the plate for higher velocities, leading to athree-dimensional fracture problem and a non-smooth crack surface.

is still uncoupled. Although the finite element code is equipped with a solver for

the conductive process, we focused this investigation on the lower velocity regime, to

verify the adequacy of our simplest model to capture crack pattern instabilities. This

assumption leave open opportunities for improvements.

Even if we are varying the temperature gap, instead of the plate width W , the

outcomes of our analysis may be compared by recourse to the dimensionless parame-

ters introduced in the section (1.3.1). In the computation of τ the two variables (W

and ∆ϑ) play the same role, with different weight.

We plot in Fig.4-9 the same results of Fig.4-1 but in terms of the Peclet number

and τ , whose expression we recall here

τ =E2(α∆ϑ)2W

Gc

(4.2)

We can observe as the extension of our data is contained within the limit of V = κ/H,

which corresponds to Pe = 1, since the validity of our model, with steady temperature

68

10−3

10−2

10−1

100

0

500

1000

1500

2000

2500

3000

Pe =HV

κ

τ=

E(α

∆θ)

2W

Gc Straight

OscillatoryBranched

Figure 4-9: Phase Field of the crack morphology, depending on the dimensionlessparameters: Pe and τ . The value Pe = 1 represents the limit of validity of the steadythermal state assumption.

profile imposed on the nodes, is restricted by the condition Pe < 1. In Fig.4-9, as

well as in Fig.4-8, we observe crack patterns almost independent of of the velocity

increments, as long as V does not get close the value of κ/H. The curves in Fig.4-8

shows a bump in correspondence of V = κ/H, which means an increment in the

fracture regularity, before establishing an opposite trend where increments in the

velocity contribute to destabilize the crack propagation. We can deduce that the

numerical results so far obtained are in good agreement with the experimental ones

available in literature.

4.2.3 Crack pattern depending on the plate width.

Referring to samples of thickness of the order of 10−1 mm, the two relevant geometric

dimensions are the plate length L and width W . The importance of the role played by

the latter has been observed in many experiments. The strain energy stored into the

material, in function of the largeness of the plate, influences the fracture propagation

as driving force.

We have chosen to investigate this aspect not varying the dimension W (so the

69

(a) Tapered plate. (b) Rectangular plate.

Figure 4-10: Fracture patterns resulting from ∆ϑ = 350 C and V = 0.001mms−1

(Cauchy stress units: N/mm2).

aspect ratio of the sample) among a certain number of experiments but performing

only a few analysis with tapered plates and examining the influence of a reducing

width on the pattern evolution. Of particular interest is the comparison of the results

with the ones obtained with a rectangular plate in the same experimental conditions.

In figure Fig.4-10a we can notice how at the bottom the growth of the fracture

is similar to the one in Fig.4-10b, even if the branch appears first in the tapered

plate than in the rectangular one. The decreasing of W implies a reduction of the

energy available to the system for feeding dissipative phenomena, in this case it

means a progressive regularization of the fracture propagation. When the limit of

energy for propagating two branches is reached, one of the cracks stops, while the one

still active evolves and kinks, disposing of extra energy to release. The path of the

second crack spontaneously tends to restore the symmetry, moving rapidly towards

the center of the plate and growing with damped oscillations. The second example

illustrates what happen to a crack growing with large oscillation when the section

of the specimen tends to reduce. We observe in Fig.4-11 how the oscillating pattern

70

(a) Tapered plate (b) Rectangular plate

Figure 4-11: Fracture patterns resulting from ∆ϑ = 300 C and V = 0.0001mms−1

(Cauchy stress units: N/mm2).

assume a sinusoidal shape and is damped by the effect of the lateral edge.

4.3 Considerations about crack tip position and

energy dissipation in the fracture process.

The fracture travels along the plate, while its tip assume a certain position respect

to the water surface. The localization of the crack tip can be included among the

parameters governing the problem.

In [33] the authors have studied the equilibrium tip position ytip as a function of

the plate width W for a fixed temperature field (∆ϑ, H and V fixed). The threshold

Wc is related to the minimal amount of energy required for crack propagation. For

larger width, i.e., inside the straight propagation region Wc < W < Wosc, the system

adapts to the excess of available elastic energy through the crack tip position ytip.

For W > Wc, the available elastic energy is greater than the quantity needed for

propagation, the crack tip position stabilizes near the hot bath, i.e., in a lower stressed

71

region. As the width decreases toward Wc , the available energy decreases, and the

tip position moves closer to the cold bath, in a higher stressed region. Finally, when

W reaches its critical value Wc , the propagation stops and the tip falls into the cold

bath. Having fixed W L above its critical Wc , the crack tip position depends on the

driving velocity and approximately follows the evolution of the thermal field with V .

It is difficult to detect the eventuality that the crack arrests and speeds up even if its

average velocity is approximately constant. The appearance of instabilities may be

characterized by a propagation discontinuous in time, with a possible high value of

the instantaneous velocity at the moment of kinking or branching.

Our model would require an higher mesh refinement to be able to discern this

aspect in detail, although we can extract some informations by comparing the plots of

the relative crack velocity arel of a straight, oscillating or branched pattern. In Fig.4-

12b the relative crack velocity is very close to the unity. In the straight propagation

the tip moves ahead of the water bath following the driving velocity. When the

fracture oscillates, the velocity of propagation increase with the wave amplitude (see

Fig.4-13b, Fig.4-14b) until becoming twice the speed of immersion in the branched

configuration of Fig.4-15b.

Analyzing the growth of the parameter τ respect to the dimensionless time tnm,

we focus in particular on the slop of the graph represented by the expression

∆τ

∆tnm=

Gca

E(α∆ϑ)2WV. (4.3)

The comparison of the value of∆τ

∆tnmfor the main crack patterns confirm that the

stable behavior corresponds to a minimum for the energy of the system. The values

of∆τ

∆tnmobtained for the straight fracture (see Fig.4-12c) are, in fact, the largest

ones. The periodically stable configuration, represented by the cracks with oscillating

morphology, is characterized by values of∆τ

∆tnmdecreasing with the wave amplitude

(see Fig.4-13c, Fig.4-14c). Finally the property of instability of the branched patterns

is supported by verifying the values of the slope of τ being the lowest ones among the

tested cases (Fig.4-15b).

72

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

V t/L

a/L

← a/V = 1.01

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

2500

V t/L

Gca

E(α

∆T)2W

L

← Gca

E(α∆T )2WV=8.5E+05

(c)

Figure 4-12: ∆ϑ = 150 C, V = 0.0001mm/s

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

V t/L

a/L

← a/V = 1.34

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

1200

1400

1600

V t/L

Gca

E(α

∆T)2W

L

← Gca

E(α∆T )2WV=6.3E+05

(c)

Figure 4-13: ∆ϑ = 200 C, V = 0.0001mms−1

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

V t/L

a/L

← a/V = 1.71

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

1200

V t/L

Gca

E(α

∆T)2W

L

← Gca

E(α∆T )2WV=4.7E+05

(c)

Figure 4-14: ∆ϑ = 300 C, V = 0.0001mms−1

73

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

V t/L

a/L

← a/V = 2.1

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600

700

800

V t/L

Gca

E(α

∆T)2W

L

← Gca

E(α∆T )2WV=3.2E+05

(c)

Figure 4-15: ∆ϑ = 350 C, V = 0.0005mm/s

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

V t/L

a/L

← a/V = 1.58

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

350

400

450

500

V t/L

Gca

E(α

∆T)2W

L

← Gca

E(α∆T )2WV=4.8E+05

(c)

Figure 4-16: ∆ϑ = 250 C, V = 0.0001mms−1, refined mesh

Within the set of results obtained imposing the temperature jump we can deduce a

trend both for the relative crack velocity and for the rate of τ . For example assuming

∆ϑ = 250 C and adopting a finer mesh the relative velocity of propagation increase

with the immersion velocity (compare Fig.4-16b, Fig.4-17b and.4-18b). At the same

time the value of∆τ

∆tnmgrows with V (compare Fig.4-16c, Fig.4-17c and.4-18c) which

means an increasing of the stability in the propagation. This results are confirmed

by the progressive regularization of the fracture pattern observed in the simulations.

74

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

V t/L

a/L

← a/V = 1.66

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600

700

V t/L

Gca

E(α

∆T)2W

L

← Gca

E(α∆T )2WV=5E+05

(c)


(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

V t/L

a/L

← a/V = 1.76

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600

700

V t/L

Gca

E(α

∆T)2W

L

← Gca

E(α∆T )2WV=5.3E+05

(c)


75

4.4 Analytical consideration on the instability man-

ifested in the crack propagation

The theory of bifurcation problems offers a framework to understand the peculiar

instability associated with a quasi-static crack propagation.

The phenomena experimentally observed can be interpreted as an Hopf bifurcation

of the solution of the equation of motion of the crack tip [36], [16], [24].

The term bifurcation was originally used by Poincare to describe the ”splitting”

of the equilibrium solutions in a family of differential equations. Given a system of

differential equations

x = fµ(x)x ∈ Rn,Rk (4.4)

depending on the k-dimensional parameter µ, the equilibrium solutions are obtained

by solving the equation

fµ(x) = 0 (4.5)

As µ varies these equilibria are described by smooth functions of µ away from the

points (x0,µ0) where the Jacobian derivative of fµ(x) with respect to x Dxfµ(x)

has a zero eigenvalue. The graph of each of these functions is a branch of equilibria,

converging at the point of bifurcation (x0,µ0).

The Hopf bifurcation refers to the development of periodic orbits (”self-oscillations”)

from a stable fixed point, as a parameter crosses a critical value. The mathematical

concept can be illustrated in a simple physical problem.

Consider the example of a rigid hoop hanging from the ceiling and a small ball

rests in the bottom (see Fig.4-19). The hoop rotates with frequency ω about a vertical

axis through its center For small values of ω, the ball stays at the bottom of the hoop

and that position is stable. However, when ω reaches some critical value ω0 , the ball

rolls up the side of the hoop to a new position y(ω), which is stable. The ball may roll

to the left or to the right, depending to which side of the vertical axis it was initially

leaning. The original stable fixed point, unique solution of the differential equations

76

Figure 4-19: Simple experiment of a hoop rotating about its vertical axis. Theposition occupied by the ball depends on the angular velocity of the rotation. When acritical value w0 is reached, the configuration with the ball at the bottom of the hoopbecomes unstable.

governing the ball’s motion, becomes unstable for ω > ω0 and splits into two stable

fixed points. For each ω > ω0 there is a stable, invariant circle of fixed points due to

the symmetries present in the problem.

In this example of a physical problem depending on a parameter, the angular

velocity the character of the solution changes abruptly and ω0 can be classified as a

Hopf bifurcation point. The appearance of the stable closed orbits is interpreted as

a ”shift of stability” from the original stationary solution to the periodic one.

Let us try to express the similarity between this example and the fracture problem

we are interested in. The drastic change in the crack morphology according to the

experimental parameters is a clear manifestation of an instability phenomenon. Fur-

thermore the arising of periodic stable solutions is evident in the oscillating patterns

characterized by wavelength and amplitude depending on the test settings (∆ϑ, V ,

W , H). The solution is conditioned by the symmetry of the problem, however, also

in this case, the first deviation from the straight direction to the left or to the right

is due to local asymmetries (material defects in the physical reality, perturbations of

77

the solution in the numerical one).

It is possible to identify another threshold, which represents the limit of stability

for the oscillating crack pattern and the appearance of branching. It delimits the

transient from periodically stable to unstable solutions.

Let us formalize the definition of a Hopf bifurcation point [24].

Suppose that the system

x = fµ(x) x ∈ Rn, µ ∈ R (4.6)

has an equilibrium point (x0, µ0) at which the following properties are satisfied:

i) Dxfµ0(x0) has a simple pair of pure imaginary eigenvalues λ(µ), λ(µ) and no other

eigenvalues with zero real parts;

ii) λ(µ), λ(µ) vary smoothly with µ along the curve of equilibria (x(µ), µ) with the

property:d

dµ(Reλ(µ)) |µ=µ0 6= 0

Under the above conditions there exist continuous functions µ = µ(ε) and T = T (ε),

depending on a parameter ε, with

µ(0) = µ0 (4.7)

T (0) = 2πβ−1 λ(µ), λ(µ) = ±iβ (4.8)

such that there are non-constant periodic solutions x(t, ε) with period T (ε) which

collapse into x0 as ε→ 0.

In [36] the authors analyze the variety of crack patterns object of this study by

the point of view of the theory of instability. They investigate the motion of the crack

tip as solution of the two basic equation describing quasi-static crack propagation

KI = KIc (4.9)

KII = 0 (4.10)

78

Figure 4-20: Reference system for the description of the crack tip position respect tothe water bath surface, denoted by the parameter s.

The position of the crack tip correspond to the point where the stress intensity fac-

tor in mode I is equivalent to its critical value and a negative KI increment for

infinitesimal crack extension. The second necessary condition is obtained assuming

the principle of local symmetry, which corresponds to a criterion of propagation of

the crack toward the direction where the shear stress vanishes.

The two equation problem has been posed adopting the hypothesis of infinite plate

width and reformulating it as a linearized eigenvalue problem. The Central Manifold

theorem guarantees that the character of the solution can be detected studying the

linear approximation of the equation, in a suitable neighborhood of the bifurcation

point.

The crack tip position has been described by two coordinates (q(s), p(s)) respect

to a reference system located at the water bath surface, marked by the variable s (see

79

Fig.4-20). The straight propagation,in particular, is obtained when p(s) = 0 holds

along all the plate. To establish the limit of stability of this solution they consider

the eigenvalue with the largest real part z∗ (in absolute value), assuming to look at

the long term behavior of the fracture (when the parameter s tends to infinity). The

sign of Re(z∗) changes accordingly with the experimental data. Of particular interest

is the point, in the data space, corresponding to a null value of Re(z∗), because it

represents the point of bifurcation of the solution, so the passage from the stable

straight solution (Re(z∗) < 0) to the instable one (Re(z∗) > 0). The Hopf bifurcation

point is characterize by a pair of pure imaginary eigenvalues (z∗ = ±iβ) and the

wavelength of the oscillating pattern is function of β.

80

Chapter 5

Conclusion

5.1 Current development

There are fundamental limits to the processing power of a single processor and in

general high resolution requires lots of memory and large computational effort. To

overcome this obstacle it is possible to distribute work and data among multiple

processors appropriately programming the sequence of instructions and the data sub-

division. SIMD (single instruction, multiple data) and MIMD (multiple instruction,

multiple data) are the traditional models for parallel machines. The memory orga-

nization can be private, for each processor, or shared. Shared memory permits to

parallelize serial program gradually, while distributed memory requires partitioning

and distributing both data and work across processors, with an initial larger effort

but with the advantage of an higher scalability.

A multi-thread version of our code, working on shared memory is already imple-

mented and it has been largely used for reducing the cost of our simulations.

We are currently working on a parallel implementation of the finite element code

adopting MPI (Message-Passing Interface), which is a message-passing library inter-

face specification base on private memory distribution.

MPI allows overlap of computation and communication, in the sense that the

computational effort is (preferably equally) distributed among all the processors and

the exchange of data happens through cooperative operations on each process.

81

MPI is not and independent language: all MPI operations are expressed as func-

tions, subroutines, or methods, according to the appropriate language bindings, which

for C, C++, Fortran-77, and Fortran-95, are part of the MPI standard. The stan-

dard has been defined through an open process by a community of parallel computing

vendors, computer scientists, and application developers. The main advantage of es-

tablishing a message-passing standard is portability.

To design parallel algorithms it is necessary to identify the part of the problem

that can be parallelized and decompose them into tasks. A crucial aspect consists of

determining the necessary communication patterns among tasks assigned to different

processors.

A good programming strategy try to maximize the work that can be done in paral-

lel, balancing the load so that it remains evenly divided along the execution. It is im-

portant to possibly reduce the work not present in the equivalent serial computation:

process startup and shutdown costs, communication, synchronization, redundancy

and speculative work.

In the parallel code under development the distribution of data is base on domain

decomposition. Each processor is delegated to perform the thermo-mechanical anal-

ysis and any topological change concerning a specific sub-domain of the whole body.

The information about forces applied on nodes shared among multiple processors is

exchanged through a suitable communication schema.

The implementation includes a re-formulation of the two-dimensional fragmenta-

tion code, able to manage the fracture propagation within each process as well as

the crossing of sub-domain boundaries and the insertion of cohesive elements along

sub-domain interfaces.

The efficiency of the parallel implementation requires minimizing the number of

messages exchanged, by reducing the inter-dependency among processors.

The code has been conceived so that it can run sequentially or in parallel, on

clusters with shared or separate memory.

82

5.2 Possible Further Developments

An immediate and natural continuation of this research may consist of including

higher immersion velocities by solving the fully coupled thermo-mechanical problem.

A first investigation of the mesh dependence issue can be carried out by intro-

ducing randomness and increasing resolution once we dispose of the complete parallel

implementation of the code. A deeper analysis of the directionality bias may involve

the study of nonlocal variational fracture methods.

We refer to generalizations of Griffith’s theory proposed in literature to deter-

mine the crack path based on energy criteria. In particular on the subject of brittle

fracture several contributions produced in the area of calculus of variation have the

attention mainly focused on the application of free discontinuity models to the pre-

diction of fracture. Free discontinuity models deal with the minimization of energy

functionals composed of bulk and surface terms, which admit the crack path as a

primary unknown. The displacement field u and its discontinuity set of Ju (repre-

senting fracture) are the arguments of an energy functional E(u, Ju) composed of

bulk potential energy and a surface (or interface) parts. The latter represents the en-

ergy dissipated during the nucleation and propagation of fractures within the body.

Optimal solutions (u, Ju) are sought through a time-continuous minimization pro-

cess. The original model in [12] dealing with a global minimization problem, has

been followed by variants considering local minimization [8]. Concerning numerical

approximations: strong approaches explicitly account for discontinuities and consider

finite element models incorporating discontinuous test functions and mesh adaptiv-

ity [26]; weak approaches model fracture through an auxiliary damage variable and

introduce energy approximation by means of families of elliptic functionals [3]. In

both cases, convergence behavior of discrete approximations has been proved, using

arguments of the Γ-convergence theory [7].

Their distinctive feature, as compared to stress based crack path tracking strate-

gies, consists of the fully variational formulation of the fracture problem, both in

terms of the displacement field and crack pattern. In [25] it has been demonstrated

83

that the convergence of free discontinuity procedures requires the adoption of very

fine fixed meshes and/or adaptive triangulations, in order to avoid mesh-dependence

of the crack predictions [13]. Adaptive models involve aspects of configuration and

mechanics, which compete against each other to determine the minimal energy solu-

tion in terms of nodal displacements, mesh geometry, and crack pattern. [39] They

may represent a rigorous approach to the mesh dependency issue, which remains open

in this work, as a stimulus for future research.

84

Appendix A

Properties and invariants of

kinematic variables

A.0.1 Index notation

So far the algebra of vector and tensor has been presented in symbolic (or direct) no-

tation but in computational mechanics it is essential to refer vectorial (and tensorial)

quantities to a basis. In order to introduce component expressions relative to a right

handed orthonormal tridimensional euclidean space we consider a set of basis vectors

e1, e2, e3 with the property of being orthonormal. Then any vector v and any tensor

A are uniquely represented by a linear combination of, rispectively, the basis vectors

or their diadic products:

v = v1e1 + v2e2 + v3e3 = vaea a = 1, 2, 3 (A.1)

A = Aij ei ⊗ ej (A.2)

where the Einstein summation convention has been adopted. We label XA, A = 1, 2, 3,

as the material (or referential) coordinates of the position vector X and xa, a = 1, 2, 3,

as the spatial (or current) coordinates of the position vector x. The deformation

gradient F is a second order tensor which involve points in two distinct configurations

85

and it is therefore called two-point tensor.

F =∂χ

∂X= Grad x(X, t) FiH =

∂xi∂XH

(A.3)

F−1 =∂χ−1

∂x= grad X(x, t) FHi =

∂XH

∂xi(A.4)

FF−1 = I FiHF−1Hj = δij (A.5)

F−1F = I F−1Hi FiK = δHK (A.6)

The expression Grad and grad are adopted to distinguish between material and spa-

tial gradient. We recall the derivatives with respect to F of the determinant of F

(Jacobian)

det F = J∂J

∂F= JF−T ∂J

∂FiH= JF−1

Hi (A.7)

The first invariant of the Cauchy-Green deformation tensor C = FTF; CHK =

FiHFiK is defined as follow:

I1 = tr C = C : I = CJJ (A.8)

and it derivative with respect to C and F are:

∂I1

∂C= I

∂CJJ∂CHK

= δJHδJK = δHK (A.9)

∂I1

∂F= 2F

∂CJJ∂FjI

=∂CJJ∂CHK

∂(FiHFiK)

∂FjI= 2FjI (A.10)

We recall the definition of the fourth-order identity tensors I and its transpost IT

(I)IJHK = δIHδJK (IT )IJHK = δIKδJH (A.11)

86

We deduce the expression of the derivative of the inverse of the deformation gradient

F−1 with respect to F:

∂F−1

∂F= −F−1 F−1 ∂F−1

Hb

∂FaK= −F−1

HaF−1Kb (A.12)

where operator for non-symmetric tensors is:

(F−1 F−1

)HbaK

= F−1HaF

−1Kb (A.13)

Proof:

F−1HbFbI = δHI

∂(F−1HbFbI

)∂FaK

=∂F−1

Hb

∂FaKFbI + F−1

HaδIK = 0 (A.14)

∂F−1Hb

∂FaKFbIF

−1Ic = −F−1

HaδIKF−1Ic

∂F−1Hb

∂FaK= −F−1

HaF−1Kb (A.15)

Or, alternatively:

FaHF−1Hb = δab

∂(FaHF

−1Hb

)∂FcK

= δacF−1Kb + FaH

∂F−1Hb

∂FcK= 0 (A.16)

F−1Ja FaH

∂F−1Hb

∂FcK= −F−1

Ja δacF−1Kb

∂F−1Jb

∂FcK= −F−1

Jc F−1Kb (A.17)

It follows that:∂F−T

∂FT=∂F−1

∂F(A.18)

and∂F−T

∂F=∂F−T

∂FT:∂FT

∂F=∂F−1

∂F: IT =

∂F−1

∂F(A.19)

87

88

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